Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      2. Extended equivariant Picard complexes and homogeneous spaces      Mikhail Borovoi  Joost van Hamel 《Transformation Groups》2012,17(1):51-86 Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

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k ) Gal\left( {{{{\bar{k}}} \left/ {k} \right.}} \right) -modules \textDiv( [`(X)] ) {\text{Div}}\left( {\bar{X}} \right) is in degree 1. We computed the isomorphism class of \textUPic( [`(G)] ) {\text{UPic}}\left( {\bar{G}} \right) in the derived category of Galois modules for a connected linear k-group G.      3. An $ \mathfrak{X} $-crown of a finite soluble group      S. F. Kamornikov  L. A. Shemetkov 《Algebra and Logic》2010,49(5):400-415 Let G be a finite soluble group and F\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which G / \textCor\texteG(M) ? \mathfrakX {{G} \left/ {{{\text{Cor}}{{\text{e}}_G}(M)}} \right.} \in \mathfrak{X} . We look at properties of a section F( G / F\mathfrakX(G) ) F\left( {{{G} \left/ {{{\Phi_\mathfrak{X}}(G)}} \right.}} \right) , which is definable for any class \mathfrakX \mathfrak{X} of primitive groups and is called an \mathfrakX \mathfrak{X} -crown of a group G. Of particular importance is the case where all groups in \mathfrakX \mathfrak{X} have equal socle length.      4. Linear Maps Preserving Operators of Local Spectral Radius Zero      Constantin Costara 《Integral Equations and Operator Theory》2012,73(1):7-16 Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let rT(x) = limsupn ? ￥ || Tnx|| 1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r T (x) = 0 if and only if rj(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r T (By) = 0 if and only if rj( T) (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.      5. Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces      Michael Fraboni  Terrence Napier 《Mathematische Zeitschrift》2010,265(3):653-685 The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}.      6. Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents      Daniel?CaroEmail author 《Mathematische Annalen》2011,349(4):747-805 Soit _boxclose{\mathcal V} un anneau de valuation discrète complet d’inégales caractéristiques (0, p), de corps résiduel parfait k, de corps des fractions K. Soient X une variété sur k, Y un ouvert de X. Nous prolongeons le théorème de pleine fidélité de Kedlaya de la manière suivante (en effet, nous ne supposons pas Y lisse): le foncteur canonique F\text-Isoc f (Y,X/K) ? F\text-Isoc f (Y,Y/K) {F\text{-}\mathrm{Isoc} ^{\dag} (Y,X/K) \to F\text{-}\mathrm{Isoc} ^{\dag} (Y,Y/K) } est pleinement fidèle. Supposons à présent Y lisse. Nous construisons la catégorie Isoc ff (Y,X/K){\mathrm{Isoc} ^{\dag\dag} (Y,X/K) } des isocristaux partiellement surcohérents sur (Y, X) dont les objets sont certains D{\mathcal D} -modules arithmétiques. De plus, nous vérifions l’équivalence de catégories sp (Y,X),+: Isoc f (Y,X/K) @ Isoc ff (Y,X/K){{\rm sp} _{(Y,X),+}: \mathrm{Isoc} ^{\dag} (Y,X/K) \cong \mathrm{Isoc} ^{\dag\dag} (Y,X/K)} .      7. Time-dependent gradient perturbations of fractional Laplacian      Tomasz Jakubowski  Karol Szczypkowski 《Journal of Evolution Equations》2010,10(2):319-339 We construct a fundamental solution of the equation ${\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0}We construct a fundamental solution of the equation ?t - Da/2 - b(·, ·) ·?x = 0{\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0} for a ? (1, 2){\alpha \in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.      8. Ricci-flat Kähler metrics on crepant resolutions of Kähler cones      Craig van Coevering 《Mathematische Annalen》2010,347(3):581-611 We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in H2c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr2 +r2 g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on crepant resolutions p:Y? X=\mathbbCn /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T n -invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in H2c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.      9. Proof of a conjecture of V. Nikiforov      Tam��s Terpai 《Combinatorica》2011,31(6):739-754 Using analytical tools, we prove that for any simple graph G on n vertices and its complement [`(G)]\bar G the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1$\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1 holds, where μ(G) and m( [`(G)] )\mu \left( {\bar G} \right) denote the greatest eigenvalue of adjacency matrix of the graphs G and [`(G)]\bar G respectively.      10. The Jacobian conjecture,the <Emphasis Type="Italic">d</Emphasis>-inversion approximation and its natural boundary      R. Peretz 《Journal of Mathematical Sciences》2010,168(3):428-436 Let F ? \mathbbC[ X, Y ]2 F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} be an étale map of degree deg F = d. An étale map G ? \mathbbC[ X,Y ]2 G \in \mathbb{C}{\left[ {X,Y} \right]^2} is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every \mathbbC2 {\mathbb{C}^2} -automorphism F of degree d has a d-inverse approximation, namely, F −1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.      11. Duality results for perturbation classes of semi-Fredholm operators      Manuel González 《Archiv der Mathematik》2011,97(4):345-352 The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC(X,Y)=PF-(X,Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ+ and PΦ− denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS(Y*,X*)=PF+(Y*,X*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC(X,Y)=PF-(X,Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC(Y*,X*)=PF-(Y*,X*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general.      12. Planar diffusions with rank-based characteristics and perturbed Tanaka equations      E. Robert Fernholz  Tomoyuki Ichiba  Ioannis Karatzas  Vilmos Prokaj 《Probability Theory and Related Fields》2013,156(1-2):343-374       13. Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions   总被引：1，自引：0，他引：1   Bin Liu  Guizhen Liu 《Graphs and Combinatorics》2010,26(6):781-791 An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠vd(G){\chi'_{vd}(G)}. It is proved that c￠vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s2(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ 2(G) denotes the minimum degree sum of two nonadjacent vertices in G.      14. Large orbits of elements centralized by a Sylow subgroup      Silvio Dolfi  Gabriel Navarro 《Archiv der Mathematik》2009,93(4):299-304 If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v1, v2 ? V{v_1, v_2 \in V} such that CG(v1)?CG(v2) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Sylp(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? CV(P){v \in {\bf C}_{V}(P)} such that |K : C K (v)|2 > |K|.      15. Geometric invariant theory and the generalized eigenvalue problem      N. Ressayre 《Inventiones Mathematicae》2010,180(2):389-441 Let G be a connected reductive subgroup of a complex connected reductive group [^(G)]\hat{G}. Fix maximal tori and Borel subgroups of G and [^(G)]{\hat{G}}. Consider the cone LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) generated by the pairs (n,[^(n)])(\nu,{\hat{\nu}}) of dominant characters such that Vn*V_{\nu}^{*} is a submodule of V[^(n)]V_{{\hat{\nu}}} (with usual notation). Here we give a minimal set of inequalities describing LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) as a part of the dominant chamber. In other words, we describe the facets of LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) which intersect the interior of the dominant chamber. We also describe smaller faces. Finally, we are interested in some classical redundant inequalities.      16. On Bernstein–Walsh-type lemmas in regions of the complex plane      F. G. Abdullayev  N.D. Aral 《Ukrainian Mathematical Journal》2011,63(3):337-350 Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition || f ||App(G): = òG | f(z) |pdsz < ￥, \left\| f \right\|_{Ap}^p(G): = \int\limits_G {{{\left| {f(z)} \right|}^p}d{\sigma_z} < \infty, }      17. The Theorem of the k-1 Happy Divorces      A.W.M. Dress 《Annals of Combinatorics》2000,4(2):183-194 Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f1, f2,...,fk: X \hookrightarrow Y X \hookrightarrow Y with # {f1(x), f2(x),...,fk(x)}=k    and    (x,f1(x)), (x, f2(x)),...,(x, fk(x)) ? R \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R for all x ? X x \in X if and only if the inequality k ·# A ￡ ?y ? Y min(k, #{a ? A | (a,y) ? R}) k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) holds for every finite subset A of X, provided {y ? Y | (x,y) ? R} \{y \in Y \mid (x,y) \in R\} is finite for all x ? X x \in X .¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every x ? X x \in X and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdös Theorem) will conclude the paper.      18. On $\frak {F}$-normal Fitting classes of finite soluble groups      S. Reifferscheid 《Archiv der Mathematik》2000,75(3):164-172 Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G\frak XG_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak Sp\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak Sp\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak Sp\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nn, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak Sp1?\frak Spr, pi \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.      19. On the existence and uniqueness of minimizers for a class of integral functionals      Graziano Crasta  Annalisa Malusa 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):129-150 We study the solvability of the minimization problem minh ? Ka ò0T a(t)[ f( |h￠(t)| ) + g( h(t) ) ]  dt,\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,      20. Simple exceptional groups of Lie type are determined by their character degrees      Hung P. Tong-Viet 《Monatshefte für Mathematik》2012,15(2):559-577 Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X1(G) í X1(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

An $ \mathfrak{X} $-crown of a finite soluble group

Let G be a finite soluble group and F_\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which G

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U^-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ^∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U^-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}.     

Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents

Soit _boxclose{\mathcal V} un anneau de valuation discrète complet d’inégales caractéristiques (0, p), de corps résiduel parfait k, de corps des fractions K. Soient X une variété sur k, Y un ouvert de X. Nous prolongeons le théorème de pleine fidélité de Kedlaya de la manière suivante (en effet, nous ne supposons pas Y lisse): le foncteur canonique F\text-Isoc ^f (Y,X/K) ? F\text-Isoc ^f (Y,Y/K) {F\text{-}\mathrm{Isoc} ^{\dag} (Y,X/K) \to F\text{-}\mathrm{Isoc} ^{\dag} (Y,Y/K) } est pleinement fidèle. Supposons à présent Y lisse. Nous construisons la catégorie Isoc ^ff (Y,X/K){\mathrm{Isoc} ^{\dag\dag} (Y,X/K) } des isocristaux partiellement surcohérents sur (Y, X) dont les objets sont certains D{\mathcal D} -modules arithmétiques. De plus, nous vérifions l’équivalence de catégories sp _(Y,X),+: Isoc ^f (Y,X/K) @ Isoc ^ff (Y,X/K){{\rm sp} _{(Y,X),+}: \mathrm{Isoc} ^{\dag} (Y,X/K) \cong \mathrm{Isoc} ^{\dag\dag} (Y,X/K)} .     

Time-dependent gradient perturbations of fractional Laplacian

We construct a fundamental solution of the equation ${\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0}We construct a fundamental solution of the equation ?_t - D^a/2 - b(·, ·) ·?_x = 0{\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0} for a ? (1, 2){\alpha \in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.     

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr² +r² g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on crepant resolutions p:Y? X=\mathbbC<sup>n</sup> /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T <sup>n</sup> -invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.     

Proof of a conjecture of V. Nikiforov

Using analytical tools, we prove that for any simple graph G on n vertices and its complement [`(G)]\bar G the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1$\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1 holds, where μ(G) and m( [`(G)] )\mu \left( {\bar G} \right) denote the greatest eigenvalue of adjacency matrix of the graphs G and [`(G)]\bar G respectively.     

The Jacobian conjecture,the <Emphasis Type="Italic">d</Emphasis>-inversion approximation and its natural boundary

Let F ? \mathbbC[ X, Y ]² F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} be an étale map of degree deg F = d. An étale map G ? \mathbbC[ X,Y ]² G \in \mathbb{C}{\left[ {X,Y} \right]^2} is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every \mathbbC² {\mathbb{C}^2} -automorphism F of degree d has a d-inverse approximation, namely, F ⁻¹. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.     

Duality results for perturbation classes of semi-Fredholm operators

The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PF₊(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC(X,Y)=PF_-(X,Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ₊ and PΦ₋ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS(Y^*,X^*)=PF₊(Y^*,X^*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC(X,Y)=PF_-(X,Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC(Y^*,X^*)=PF_-(Y^*,X^*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS(X,Y)=PF₊(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general.     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.     

Large orbits of elements centralized by a Sylow subgroup

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v₁, v₂ ? V{v_1, v_2 \in V} such that C_G(v₁)?C_G(v₂) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Syl_p(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? C_V(P){v \in {\bf C}_{V}(P)} such that |K : C _K (v)|² > |K|.     

Geometric invariant theory and the generalized eigenvalue problem

Let G be a connected reductive subgroup of a complex connected reductive group [^(G)]\hat{G}. Fix maximal tori and Borel subgroups of G and [^(G)]{\hat{G}}. Consider the cone LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) generated by the pairs (n,[^(n)])(\nu,{\hat{\nu}}) of dominant characters such that V_n^*V_{\nu}^{*} is a submodule of V_[^(n)]V_{{\hat{\nu}}} (with usual notation). Here we give a minimal set of inequalities describing LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) as a part of the dominant chamber. In other words, we describe the facets of LR(G,[^(G)])\mathcal{LR}(G,{\hat{G}}) which intersect the interior of the dominant chamber. We also describe smaller faces. Finally, we are interested in some classical redundant inequalities.     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

The Theorem of the k-1 Happy Divorces

Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f₁, f₂,...,f_k: X \hookrightarrow Y X \hookrightarrow Y with # {f₁(x), f₂(x),...,f_k(x)}=k    and    (x,f₁(x)), (x, f₂(x)),...,(x, f_k(x)) ? R \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R for all x ? X x \in X if and only if the inequality k ·# A ￡ ?_{y ? Y} min(k, #{a ? A | (a,y) ? R}) k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) holds for every finite subset A of X, provided {y ? Y | (x,y) ? R} \{y \in Y \mid (x,y) \in R\} is finite for all x ? X x \in X .¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every x ? X x \in X and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdös Theorem) will conclude the paper.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

On the existence and uniqueness of minimizers for a class of integral functionals

We study the solvability of the minimization problem

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.